Question

A technician is working with dry ice (solid \(\mathrm{CO}_{2}\) ) in a closed 6000 liter fume hood maintained at a constant pressure of \(1.00 \mathrm{~atm}\) and a constant temperature of \(27^{\circ} \mathrm{C}\). If he has not been conditioned to tolerate \(\mathrm{CO}_{2}\), he will succumb to \(\mathrm{CO}_{2}\) poisoning when the concentration rises to \(10 \%\), by volume, of the atmosphere. If the ventilation stops, what minimum weight of dry ice must sublimate to constitute a hazard? Assume that there was no \(\mathrm{CO}_{2}\) present initially.

Short answer

The minimum weight of dry ice that must sublimate to constitute a hazard is approximately \(1076.19\, \mathrm{g}\).

Step-by-step solution

Convert the temperature to Kelvin

First, we must convert the given temperature from Celsius to Kelvin. We do this by adding 273.15 to the Celsius temperature. \(T(K) = 27^{\circ} \mathrm{C} + 273.15\) \(T(K) = 300.15\, \mathrm{K}\)

Find the total moles of gas in the fume hood

Now that we have the temperature in Kelvin, we can use the ideal gas law to find the total moles of gas in the fume hood. The ideal gas law is: \(PV = nRT\) where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. We will solve for the total moles of gas (n) in the fume hood. \(n = \frac{PV}{RT}\) \(n = \frac{(1.00\, \mathrm{atm})(6000\, \mathrm{L})}{(0.0821\, \mathrm{\frac{L\cdot atm}{mol\cdot K}})(300.15\, \mathrm{K})}\) \(n = 244.57\, \mathrm{moles}\)

Determine the moles of CO₂ needed to reach a \(10\%\) concentration

We need to find the moles of CO₂ required so that the concentration of CO₂ in the atmosphere reaches \(10\%\). We can do that by multiplying the total moles of gas by the desired concentration. \(n_{\mathrm{CO_2}} = 244.57\, \mathrm{moles} \times 0.10\) \(n_{\mathrm{CO_2}} = 24.46\, \mathrm{moles}\)

Convert the moles of CO₂ to weight

Now that we have the moles of CO₂ needed, we can finally convert it to weight. The molecular weight of CO₂ is 44.01 g/mol. We will multiply the moles of CO₂ by this molecular weight: Weight of CO₂ = moles of CO₂ × molecular weight of CO₂ Weight of CO₂ = \(24.46\, \mathrm{moles}\) × \(44.01\, \mathrm{g/mol}\) Weight of CO₂ = \(1076.19\, \mathrm{g}\) So, the minimum weight of dry ice that must sublimate to constitute a hazard is approximately 1076.19 g.

Key concepts

Ideal Gas Law

The ideal gas law is a crucial equation in chemistry and physics that relates the pressure, volume, temperature, and number of moles of a gas. It is usually expressed as \(PV = nRT\), where \(P\) stands for pressure, \(V\) for volume, \(n\) for the number of moles, \(R\) for the ideal gas constant, and \(T\) for the temperature in Kelvin. In practical applications, such as the dry ice sublimation problem, it allows us to calculate how a change in one variable, such as the sublimation of dry ice increasing the number of moles of gas, might affect the others in a closed system.

The ideal gas constant \(R\) has a value of \(0.0821 \frac{L\cdot atm}{mol\cdot K}\), and it plays a pivotal role in bridging the units of pressure, volume, and temperature. The assumed condition is that gaseous substances behave ‘ideally’ or perfectly, which may not always be the case in real-world scenarios, but provides a good approximation under standard conditions. When working through the dry ice exercise, the understanding of how to manipulate this law is fundamental in determining the risk of CO2 poisoning within the 6000 liter fume hood.

Gas Volume and Molarity

When we refer to the volume of a gas, we are considering the three-dimensional space that the gas occupies, measured in liters (L) in the context of chemistry. Molarity, meanwhile, is a measure of the concentration of a substance in a solution and is defined as moles of solute per liter of solution (mol/L).

While molarity is generally used for solutions, the concept can be applied to gases when addressing their concentration in a mixture, such as air in a room or a fume hood. In the dry ice sublimation calculation, for example, we are concerned with the molarity of CO2 in the air to determine when the concentration reaches a hazardous level. To find this, we calculate the volume that the CO2 would occupy at a certain molarity and then translate this volume into a percentage of the total air volume. Knowing this percentage tells us about the concentration of CO2 and whether it has reached a potentially toxic level. This is vital for ensuring the safety of individuals who may be exposed to the environment where the gas is present.

Molecular Weight Calculation

Molecular weight calculation is a fundamental concept in chemistry that entails adding the atomic masses of each element in a molecule. The molecular weight, which is also known as the molar mass, is typically expressed in grams per mole (g/mol). This calculation allows us to convert between the mass of a substance and the number of moles, a key step in many stoichiometric calculations including gas law problems.

For CO2, which is composed of one carbon atom and two oxygen atoms, the molecular weight can be calculated by summing the atomic masses of these atoms based on the periodic table (approximately 12.01 g/mol for Carbon and 16.00 g/mol for Oxygen). Consequently, the molecular weight for CO2 would be \((1 \times 12.01) + (2 \times 16.00) = 44.01\ g/mol\). When we know the number of moles of CO2 needed to reach a dangerous level, as determined by the ideal gas law, we can use the molecular weight of CO2 to find out the corresponding mass of dry ice needing to sublimate to reach that level, completing the solution to the exercise.